Copied to
clipboard

G = D28:5D4order 448 = 26·7

5th semidirect product of D28 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D28:5D4, C42:6D14, Dic14:5D4, (C2xD4):2D14, C4:1D4:4D7, C4.55(D4xD7), C7:3(D4:4D4), C28.35(C2xD4), D4:6D14:4C2, (C4xC28):14C22, C28.D4:6C2, (D4xC14):2C22, Dic14:C4:13C2, C14.53C22wrC2, D4.D14:3C2, (C22xC14).23D4, C4.Dic7:7C22, (C2xC28).395C23, C4oD28.21C22, C23.11(C7:D4), C2.21(C23:D14), (C7xC4:1D4):4C2, (C2xC14).526(C2xD4), C22.33(C2xC7:D4), (C2xC4).118(C22xD7), SmallGroup(448,611)

Series: Derived Chief Lower central Upper central

C1C2xC28 — D28:5D4
C1C7C14C28C2xC28C4oD28D4:6D14 — D28:5D4
C7C14C2xC28 — D28:5D4
C1C2C2xC4C4:1D4

Generators and relations for D28:5D4
 G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, ac=ca, dad=a15, cbc-1=a21b, dbd=a14b, dcd=c-1 >

Subgroups: 940 in 168 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2xC4, C2xC4, D4, Q8, C23, C23, D7, C14, C14, C42, M4(2), D8, SD16, C2xD4, C2xD4, C4oD4, Dic7, C28, C28, D14, C2xC14, C2xC14, C4.D4, C4wrC2, C4:1D4, C8:C22, 2+ 1+4, C7:C8, Dic14, C4xD7, D28, C2xDic7, C7:D4, C2xC28, C2xC28, C7xD4, C22xD7, C22xC14, C22xC14, D4:4D4, C4.Dic7, D4:D7, D4.D7, C4xC28, C4oD28, D4xD7, D4:2D7, C2xC7:D4, D4xC14, D4xC14, Dic14:C4, C28.D4, D4.D14, C7xC4:1D4, D4:6D14, D28:5D4
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, D14, C22wrC2, C7:D4, C22xD7, D4:4D4, D4xD7, C2xC7:D4, C23:D14, D28:5D4

Smallest permutation representation of D28:5D4
On 56 points
Generators in S56
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)
(1 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)
(29 50 43 36)(30 51 44 37)(31 52 45 38)(32 53 46 39)(33 54 47 40)(34 55 48 41)(35 56 49 42)
(1 8)(2 23)(3 10)(4 25)(5 12)(6 27)(7 14)(9 16)(11 18)(13 20)(15 22)(17 24)(19 26)(21 28)(29 50)(30 37)(31 52)(32 39)(33 54)(34 41)(35 56)(36 43)(38 45)(40 47)(42 49)(44 51)(46 53)(48 55)

G:=sub<Sym(56)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49), (29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42), (1,8)(2,23)(3,10)(4,25)(5,12)(6,27)(7,14)(9,16)(11,18)(13,20)(15,22)(17,24)(19,26)(21,28)(29,50)(30,37)(31,52)(32,39)(33,54)(34,41)(35,56)(36,43)(38,45)(40,47)(42,49)(44,51)(46,53)(48,55)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49), (29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42), (1,8)(2,23)(3,10)(4,25)(5,12)(6,27)(7,14)(9,16)(11,18)(13,20)(15,22)(17,24)(19,26)(21,28)(29,50)(30,37)(31,52)(32,39)(33,54)(34,41)(35,56)(36,43)(38,45)(40,47)(42,49)(44,51)(46,53)(48,55) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,30),(20,29),(21,56),(22,55),(23,54),(24,53),(25,52),(26,51),(27,50),(28,49)], [(29,50,43,36),(30,51,44,37),(31,52,45,38),(32,53,46,39),(33,54,47,40),(34,55,48,41),(35,56,49,42)], [(1,8),(2,23),(3,10),(4,25),(5,12),(6,27),(7,14),(9,16),(11,18),(13,20),(15,22),(17,24),(19,26),(21,28),(29,50),(30,37),(31,52),(32,39),(33,54),(34,41),(35,56),(36,43),(38,45),(40,47),(42,49),(44,51),(46,53),(48,55)]])

58 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F7A7B7C8A8B14A···14I14J···14U28A···28R
order122222224444447778814···1414···1428···28
size11244828282244282822256562···28···84···4

58 irreducible representations

dim1111112222222444
type++++++++++++++
imageC1C2C2C2C2C2D4D4D4D7D14D14C7:D4D4:4D4D4xD7D28:5D4
kernelD28:5D4Dic14:C4C28.D4D4.D14C7xC4:1D4D4:6D14Dic14D28C22xC14C4:1D4C42C2xD4C23C7C4C1
# reps121211222336122612

Matrix representation of D28:5D4 in GL4(F113) generated by

01600
97000
000106
0070
,
0070
000106
97000
01600
,
1000
0100
0001
001120
,
0100
1000
0001
0010
G:=sub<GL(4,GF(113))| [0,97,0,0,16,0,0,0,0,0,0,7,0,0,106,0],[0,0,97,0,0,0,0,16,7,0,0,0,0,106,0,0],[1,0,0,0,0,1,0,0,0,0,0,112,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

D28:5D4 in GAP, Magma, Sage, TeX

D_{28}\rtimes_5D_4
% in TeX

G:=Group("D28:5D4");
// GroupNames label

G:=SmallGroup(448,611);
// by ID

G=gap.SmallGroup(448,611);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,1123,570,297,136,1684,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^15,c*b*c^-1=a^21*b,d*b*d=a^14*b,d*c*d=c^-1>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<